The Role of Mathematics in Physics

It took a long time in the history of humankind before it occurred to
anyone that mathematics is useful - even vital - in the understanding
of nature. Western thought was dominated from antiquity to the
Renaissance, turn by turn, by Plato and Aristotle. Plato taught that
reality consists of idealized "forms", and our world was a flawed,
inadequate shadow of reality - hardly worthy of passing notice, let
alone study. Aristotle thought that the intricacies of nature could
never be described by the abstract simplicity of
mathematics.1 Galileo recognized and used the power of
mathematics in his study of nature, and with his discovery modern
science was born.

In the text
(section 1.2 Mathematics - The Language of Science, page 1),
Professor Hewitt discusses some of the roles that mathematics plays
in science, particularly physics - as well as why mathematics is
not emphasized in this particular physics course.

A couple of points about the discussion in the book:

While it is true that most scientists would agree with Prof.
Hewitt's claim that "when the ideas of science are expressed
mathematical terms, they are unambiguous" (page 1), some would
object that a mathematical statement can't be more precise than
the (verbal) concepts and definitions that it came from.

You should understand that while the statement, "When the
findings in nature are expressed mathematically, they are easier
to verify or disprove by experiment" (also page 1) is certainly
true, Prof. Hewitt is not saying that it is
easy to "verify or disprove" something expressed
mathematically by experiment - because it isn't (as you
will see)! Most students' experience with experiment involves
following a fill-in-the-blanks lab book, and then writing the
conclusion "It didn't work because of experimental error,"
turning it in, and forgetting about it - or some such baloney. In
reality, it requires a great deal of thought, skill, and
perseverance to get even reasonable experimental results.

Mathematics as Abbreviation:

A role that mathematics plays in physics not mentioned in the text
is that mathematics is a really great way to get a very concise
statement that would take a lot of words in English. For example,
Newton's Second Law
can be stated as follows:

The magnitude of the acceleration of an object is
directly proportional to the net force applied to the object, and
inversely proportional to the object's mass. The direction of the
acceleration is the same as the direction of the net force.

Exactly what all of this means is not important (at the moment) -
what is important is that the statement above can be expressed
mathematically as:

The point is that to a physicist, both statements say
exactly the same thing. The symbolism of mathematics can
replace a lot of words with just a few symbols.

Mathematics as Concept Map:

Many beginning physicists get the notion that equations in physics
are just something to "plug the numbers into and get the answer" -
which is one reason that numerical calculation is not emphasized in
this physics course. Physicists think differently - equations tell
them how concepts are linked together.

For instance, this equation arises in the study of kinematics:

The symbol on the left side of the equation represents the concept
"average velocity". Since there are two symbols (forgetting the
division sign, and the
counts as one symbol) on the right side, to a physicist, the equation
says (among other things) that the average velocity of an object
depends on two (and only two) other concepts - the object's
displacement (),
and the time it has been moving (t). Thus equations tell scientists
how concepts are related to one another.

Mathematics as Mechanized
Thinking:

Once an idea is expressed in mathematical form, you can use the
rules (axioms, theorems, etc.) of mathematics to change it into other
statements. If the original statement is correct, and you follow the
rules faithfully, your final statement will also be correct. This is
what you do when you "solve" a mathematics problem.

From a scientific point of view, however, if you start with one
statement about nature, and end up with another statement about
nature, what you have been doing is thinking about nature.
Mathematics mechanizes thinking. That's why you use it to solve
problems! You could (possibly) figure it out without the help of
mathematics, but mathematics makes it so much easier because all you
have to do is follow the rules!

As a very simple example, suppose you start with the equation
above, which is often considered to be the definition of average
velocity (in mathematical form, of course):

It is a perfectly acceptable mathematical operation to multiply
both sides of an equation by a variable, so multiply both sides of
this equation by "t". You get:

On the right side, the rules of algebra say that t/t = 1, so it
must be true that:

And the commutative property of algebra says that this is the same
as:

This is a new statement about nature (equivalent to the familiar
"distance equals speed times time") - derived using the rules of
mathematics. Using mathematics, physicists can discover new
relationships among physical quantities - mathematics mechanizes
thinking.